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TWMS Jour. Pure Appl. Math., V.2, N.2, 2011, pp.256-270

A NEW METHOD FOR RANKING FUZZY NUMBERS WITHOUTCONCERNING OF REAL NUMBERS

R. EZZATI1, R. ENAYATI1, A. MOTTAGHI1, R. SANEIFARD2

Abstract. In this paper, a new ranking function is proposed to compare m fuzzy quantities

in m− 1 steps. The ranking function is in terms of the core and margin and α− cuts of fuzzy

quantity, which are considered simultaneously. Also, in our procedure the real numbers will be

omitted. So, we have illustrated several typical examples to compare the proposed method with

some other ranking methods.

Keywords: fuzzy numbers, ranking, the total difference.

AMS Subject Classification: 03B52.

1. Introduction

In a fuzzy environment, ranking fuzzy numbers is a very important decision making pro-cedure. Many authors have investigated various ranking methods. Wang and Kerre [23, 24]organized more than thirty ordering indices and classified them into three categories, fuzzymean and spread, fuzzy scoring and preference relation. In the first class, each index is associ-ated with a mapping from the set of fuzzy quantities to the real line in order to transform theinvolved fuzzy quantities into real numbers. Fuzzy quantities are then compared according tothe order of corresponding real numbers. And it has only been referenced to the first articleof these authors. Other contributions in this field are: Ranking fuzzy numbers based on fuzzysimulation analysis method proposed by Huijun Sun and Jianjun Wu [22], The revised methodof ranking fuzzy numbers with an area between the centroid and original points proposed by Yu-Jie Wang and Hsuan-Shih Lee [27], Ranking fuzzy numbers by distance minimization proposedby B. Asady and A. Zendehnam [3], Ranking fuzzy numbers with preference weighting functionexpectations proposed by Xin-Wang Liu and Shi-Lian Han [20], Ranking fuzzy numbers with anarea method using radius of gyration proposed by Yong Deng, Zhu Zhenfu and Liu Qi [15], Anapproximate approach for ranking fuzzy numbers based on left and right dominance proposed byLiang-Hsuan Chen and Hai-Wen Lu [7], The revised method of ranking LR fuzzy number basedon deviation degree proposed by B. Asady [4], A new approach for ranking of trapezoidal byfuzzy numbers proposed by S. Abbasbandy and T. Hajjari [1], Ranking LR fuzzy number basedon deviation degree proposed by Zhong-Xing Wanga, Yong-Jun Liua, Zhi-Ping Fanb and BoFeng [21], Area ranking of fuzzy numbers based on positive and negative ideal points proposedby Ying-Ming Wanga and Ying Luob [25], Ranking of fuzzy numbers by sign distance proposedBy S. Abbasbandy and B. Asady [2].

1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran,

e-mail: [email protected] Department of Mathematics, Oroumieh Branch, Islamic Azad University, Oroumieh, Iran

Manuscript received January 2011.

256

R. EZZATI, R. ENAYATI, A. MOTTAGHI, R. SANEIFARD: A NEW METHOD FOR RANKING ... 257

Bortolan and Degani [6] were the first authors who made a review of the literature andprovided not only a systematical review of fuzzy ranking research but also gave results of com-parisons among various ranking methods, checking the disparity of results among them. In theBortolan and Degani’s paper, there are examples which have been indicated in the paper likeFig. 3. Set 3.

Chen and Hwang [8] were the first author, who presented twenty ranking methods classifiedinto three major classes according to the media that each method uses. There are preferencerelation method, a fuzzy mean and spread method, fuzzy scoring method, and linguistic method.

So Liou and Wang [19] proposed a method for ranking fuzzy numbers with an integral valuethat represents a ”mean value”, and there are examples such as those used in Fig. 5. Set 2. AlsoThis method widely used the method in literature which were modified by Garcia-Cascales andLamata [16] is able to discriminate in cases such as Fig. 4. Set 1 or Fig. 8. Set 3, while manyother methods were not capable. It is very interesting that our method is able to discriminatein cases such as Fig. 7. Set 2 while no method was able to discriminate different types of fuzzynumbers as in Fig. 6. Set 1.

In proposed method, the core and margin and α − cuts are considered according to each offuzzy quantities simultaneously. Also there are some properties which are useful in ranking alarge quantity of fuzzy numbers in this described method. It is clear that the proposed method,with respect to these properties, is more reliable and computationally easier than some methodswhich were proposed up to now and overcome the shortcoming of the previous methods. Forexample, comparing the proposed method with the existing method Liang-Hsuan Chen and Hai-Wen Lu [7] reveals that the proposed method is more simple and consistent than the proposedmethod in [7].

The rest of this paper is organized as follows. Section 2 briefly introduces the fuzzy num-bers. Section 3 introduces the ranking approach. Section 4 describes some useful properties.Concluding remarks are finally made in Section 5.

2. Basic definitions and notations

A real fuzzy number can be defined as a fuzzy subset of the real line <, which is convexand normal. That is, for a fuzzy number A of < defined by membership function µA(x), x ∈<,the following relations exist:

maxx

µA(x) = 1, (1)

µA[λx1 + (1− λ)x2] ≥ min[µA(x1), µA(x2)], (2)

where x1, x2 ∈ <,∀λ ∈ [0, 1]. A fuzzy number A with the membership function µA(x), x ∈ <,can be defined as follows:

µA(x) =

µLA(x), a ≤ x ≤ b,

1, b ≤ x ≤ c,

µRA(x), c ≤ x ≤ d,

0, otherwise,

(3)

where µLA(x) is the left membership function that is an increasing function and µL

A(x) : [a, b] →[0, 1]. Meanwhile, µR

A(x) is the right membership function that is a decreasing function andµR

A(x) : [c, d] → [0, 1]. In addition, a trapezoidal fuzzy number denoted by [a, b, c, d] can alsosignify a triangular fuzzy number if b = c. Assume that, every fuzzy number is bounded; i.e.−∞ < a, d < ∞. The support of fuzzy number A is denoted as supp(A) and obtained by{x ∈ <|µA(x) > 0} and the core(A) leads to a set such that {x ∈ <|µA(x) = 1}.

258 TWMS JOUR. PURE APPL. MATH., V.2, N.2, 2011

The α− cuts (level sets) of fuzzy number A can be obtained as follows:

A(α) = {x ∈ <|µA(x) ≥ α}, α ∈ [0, 1], (4)

where A(α) is a convex subset that belongs to <. The lower and upper limits of α-cut Ai aredefined as:

Ai(α) = inf{x|µAi(x) ≥ α}, (5)

Ai(α) = sup{x|µAi(x) ≥ α}, (6)

where Ai(1) − Ai(0) and Ai(0) − Ai(1) are left and right spreads for each fuzzy number Ai,respectively.

3. The total difference between fuzzy numbers

In this section, an operator will be proposed as D(A,B) and called the total differencebetween the fuzzy numbers. To compare A and B, we need to determine some parameters asfollows:

MA = sup{α : µA(x) = α, ∀ x ∈ supp(A)}. (7)

So MA = 1, if A be a fuzzy number.

core(A) = sup{x : µA(x) = MA}, (8)

core(A) = inf{x : µA(x) = MA}, (9)

If A be a fuzzy number such that supp(A) = [a, b], we define:

supp(A) = max{x : x ∈ supp(A)} = b, (10)

supp(A) = min{x : x ∈ supp(A)} = a. (11)

So, the total difference of A and B can be organized as:

D(A,B) = (core(A) + core(A)

suppA− supp(A))2

∫ MA

0(A(r)−A(r)) dr− (12)

−(core(B) + core(B)

supp(B)− supp(B))2

∫ MB

0(B(r)−B(r)) dr.

In sum, the algorithm to rank of two fuzzy numbers A and B based on total difference betweenthem is given as follows:Step 1. Let supp(A) = [a1, b1] , supp(B) = [a2, b2], and C = min{a1, a2}. If C be nonnegative,go to step 2. Otherwise, add the value of |C| to all fuzzy numbers (So, all fuzzy numbers aremoved into the nonnegative x-axis ), and then go to step 2.Step 2. Determine the total deference between A and B by applying (3.11), i.e. D(A, B), andthen go to step 3.Step 3. The ranking order is determined based on the following rules:1) If D(A,B) > 0, then A Â B,

2) If D(A,B) < 0, then A ≺ B,

3) If D(A,B) = 0 and supp(A) > supp(B) (i.e. b1 > b2), then A Â B,4) If D(B, A) = 0 and supp(B) > supp(A) (i.e. b2 > b1), then B Â A, otherwise A ∼ B.

Note that in this method if we encounter fuzzy numbers such that a part of their supportor all of their support lies on negative x-axis, we will first obtain minimum of support amongall them i.e. k=min{Ai(0), i = 1, 2, ..., n}, and then we will add the absolute value of obtainedvalue to all fuzzy numbers i.e. A = [a, b, c, d] transforms into A = [a+ |k|, b+ |k|, c+ |k|, d+ |k|].


4. Numerical examples

In this section, the researchers compare the proposed method with the others in [5, 9, 11, 12,13, 14, 29].

Example 4.1. Consider the following sets, see Yao and Wu [29].Set 1: A=(0.4,0.5,1), B=(0.4,0.7,1), C=(0.4,0.9,1).Set 2: A=(0.3,0.4,0.7,0.9) (trapezoidal fuzzy number) , B=(0.3,0.7,0.9), C=(0.5,0.7,0.9).Set 3: A=(0.3,0.5,0.8,0.9) (trapezoidal fuzzy number), B=(0.2,0.5,0.9), C=(0.1,0.6,0.8).To compare other methods, researchers refer reader to Table 1.

0.4 0.5 0.7 0.9 10

1

CBA

Fig. 1. Set 1.

0.3 0.4 0.5 0.7 0.90

1

A B C

Fig. 2. Set 2.

0.1 0.2 0.3 0.5 0.6 0.8 0.90

1

AB C

Fig. 3. Set 3.


Table 1. Comparative results of example 4.1.Authors Fuzzy Set1 Set2 Set3

number

Proposed method A,B -0.8 -0.123 1.401B,C -1.07 -0.815 -0.215A,C -1.87 -0.938 1.716

Results A ≺ B ≺ C A ≺ B ≺ C B ≺ C ≺ A

Sing Distance method A 1.2000 1.1500 0.0950with p=1 B 1.4000 1.3000 1.0500

C 1.6000 1.4000 1.0500Results A ≺ B ≺ C A ≺ B ≺ C A ≺ B ∼ C

Sing Distance method A 0.8869 0.8756 0.7853with p=2 B 1.0194 0.9522 0.7958

C 1.1605 1.0033 0.8386Results A ≺ B ≺ C A ≺ B ≺ C A ≺ B ≺ C

Distance Minimization A 0.6 0.575 0.475B 0.7 0.65 0.525C 0.9 0.7 0.525

Result A ≺ B ≺ C A ≺ B ≺ C A ≺ B ∼ C

Abbasbandy and Hajjari A 0.5334 0.5584 0.5250(Magnitude method) B 0.7000 0.6334 0.5084

C 0.8666 0.7000 0.5750Result A ≺ B ≺ C A ≺ B ≺ C B ≺ A ≺ C

Choobineh and Li A 0.3333 0.5480 0.5000B 0.5000 0.5830 0.5833C 0.6670 0.6670 0.6111

Results A ≺ B ≺ C A ≺ B ≺ C A ≺ B ≺ C

Yager A 0.6000 0.5750 0.4500B 0.7000 0.6500 0.5250C 0.8000 0.7000 0.5500


Chen A 0.3375 0.4315 0.5200B 0.5000 0.5625 0.5700C 0.6670 0.6250 0.6250


Baldwin and Guild A 0.3000 0.2700 0.4000B 0.3300 0.2700 0.4200C 0.4400 0.3700 0.4200

Results A ≺ B ≺ C A ∼ B ≺ C A ≺ B ∼ C

Chu and Tsao A 0.2990 0.2847 0.2440B 0.3500 0.3247 0.2624C 0.3993 0.3500 0.2619

Results A ≺ B ≺ C A ≺ B ≺ C A ≺ C ≺ B

Yao and Wu A 0.6000 0.5750 0.4750B 0.7000 0.6500 0.5250C 0.8000 0.7000 0.5250

Results A ≺ B ≺ C A ≺ B ≺ C A ≺ B ∼ C

Cheng distance A 0.7900 0.7577 0.7106B 0.8602 0.8149 0.7256C 0.9268 0.8602 0.7241

Results A ≺ B ≺ C A ≺ B ≺ C A ≺ C ≺ B


Table 1. (continued)Authors Fuzzy Set1 Set2 Set3

number

Cheng CV uniform A 0.0272 0.0328 0.0693distribution B 0.0214 0.0246 0.0385

C 0.0225 0.0095 0.0433Results A ≺ C ≺ B A ≺ B ≺ C A ≺ C ≺ B

Cheng CV proportional A 0.0183 0.0260 0.0471distribution B 0.0128 0.0146 0.0236

C 0.0137 0.0057 0.0255Results A ≺ C ≺ B A ≺ B ≺ C A ≺ C ≺ B

Note that, in Table 1 and in set 3, for Sign Distance (p = 1), Distance Minimization, Chu-Tsao and Yao-Wu methods, the ranking order for fuzzy numbers B and C is B ∼ C, that looksunreasonable.Example 4.2. Consider the following sets:Set 1: A = (1, 2, 5), B = (0, 3, 4) and C = (2, 2.5, 3).Set 2: A = (3, 5, 7; 1), B = (3, 5, 7; 0.8), C = (5, 7, 9, 10), D = (6, 7, 9, 10; 0.6) and E =(7, 8, 9, 10; 0.4).To compare some of the other methods in [14], the readers can refer to Table 2.

0 1 2 2.5 3 4 50

1

A BC

Fig. 4. Set 1.

3 5 6 7 8 9 100

0.4

0.6

0.8

1

AC

D

E

B

Fig. 5. Set 2.


Table 2. Comparative results of Example 4.2.Proposed Sign distance Distance Chu and Tsao CV Magnitude

FNs method p=2 Minimization (Revisited) index method

Set 1.A D(A,B) = −2.5 3.9157 2.5 0.74 0.32 2.16B D(B, C) = −8 3.9157 2.5 0.74 0.36 2.83C D(A,C) = −10.5 3.5590 2.5 0.75 0.08 2.5

Results A ≺ B ≺ C C ≺ A ∼ B C ∼ A ∼ B A ∼ B ≺ C B ≺ A ≺ C A ≺ C ≺ B

Set 2.A D(A,B) = 7.5 7.25 5.00 2.50 0.16 5.00B D(B, C) = −25.5 6.49 5.00 2.00 0.16 5.00C D(C,D) = 7.2 11.54 7.75 3.89 0.14 7.91D D(D,E) = 3.12 8.92 8.00 2.40 0.10 8.00E D(A,E) = −13.8 7.65 8.00 1.70 0.06 8.50

B ≺ A ≺ E B ≺ A ≺ E A ∼ B ≺ C E ≺ B ≺ D E ≺ D ≺ C A ∼ B ≺ C

E ≺ D ≺ C E ≺ D ≺ C C ≺ D ∼ E D ≺ A ≺ C C ≺ A ∼ B C ≺ D ≺ E

In Liou and Wang [19] ranking method, there was A ∼ B ≺ C ≺ D ≺ E. In Chu and Tsao [14]ranking method, there was E ∼ B ≺ D ≺ A ≺ C. In Ching-Hsue Cheng [11] ranking method,there was B ≺ A ≺ C ≺ D ≺ E. By using proposed method, there is B ≺ A ≺ E ≺ D ≺ C.From Fig. 5, the researchers can conclude that this order is more consistent with human intu-ition.Example 4.3. Consider the following sets:Set 1: A1 = (0.6, 0.7, 0.8), A2 = (0.4, 0.5, 0.6, 0.7), A3 = (0.2, 0.5, 0.8),A4 = (0.3, 0.4, 0.9), A5 = (0.1, 0.2, 0.3).Set 2: B1 = (0.1, 0.3, 0.5), B2 = (0.2, 0.3, 0.4).Set 3: C1 = (0.2, 0.5, 0.8), C2 = (0.3, 0.4, 0.9).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1A3 A2 A1

A4A5

Fig. 6. Set 1.


0.1 0.2 0.3 0.4 0.50

1

B1B2

Fig. 7. Set 2.

0.2 0.3 0.4 0.5 0.8 0.90

1C1C2

Fig. 8. Set 3.

In Chen and Lu [7] ranking method, there were A1 Â A2 Â A3 ∼ A4 Â A5, when β = 0.0 andA1 Â A2 Â A3 ∼ A4 Â A5, when β = 0.5 and A1 Â A2 ∼ A3 ∼ A4 Â A5, when β = 1.0. Byusing proposed method, the ranking order is A1 Â A2 Â A3 Â A4 Â A5. From Fig. 6, we canconclude that this order is more consistent with human intuition.

In Chu and Tsao [7], Asady and Zendehnam [3], Yager [28], Wang centroid method [26] andBaldwin and Guild [5] ranking methods, there were B1 ∼ B2. In Chen [9] ranking method, therewas B1 ≺ B2. By using proposed method, there is B1 ≺ B2. From Fig. 7, we can conclude thatB1 ≺ B2 is more consistent with human intuition.

Table 3. Comparative results of Example 4.3.Authors Set1 Set2 Set3

Proposed method D(A1, A3) = 4.07 D(B1, B2) = −0.45 D(C1, C2) = 0.3D(A2, A3) = 1.85D(A4, A3) = −0.3D(A5, A3) = −0.43A1 Â A2 Â A3 Â A4 Â A5 B2 Â B1 C1 Â C2

Example 4.4. In this example, we investigate special types of fuzzy number and rank them.Set 1: H1 = (1, 2, 2),H2 = (2, 2, 3) (the fuzzy numbers are symmetric and positive, also theirsupport have equal length ).Set 2: K1 = (−3,−1, 1, 3),K2 = (−2,−1, 1, 2) ( two symmetric trapezoidal fuzzy numbers such


that a part of their supports are negative ).Set 3: L1 = (−4,−3,−1, 0), L2 = (−3,−2,−1) ( the negative fuzzy number ).

1 2 30

1

H1 H2

Fig. 9. Set 1.

−3 −2 −1 0 1 2 30

1

K1 K2

Fig. 10. Set 2.

−4 −3 −2 −1 00

1

L1 L2

Fig. 11. Set 3.

It is clear that because of D(H1,H2) = 0 and supp(H2) > supp(H1), we have H2 Â H1. Also,for ranking set2 and set3, we first move all of the fuzzy numbers with values | − 3| and | − 4| toright side of x-axis, respectively, and then we rank all of them.


0 1 2 3 4 5 60

1

K1 K2

Fig. 12.

0 1 2 3 40

1

L1 L2

Fig. 13.

Table 4. Comparative results of Example 4.4.Authors Set1 Set2 Set3

Proposed method D(H1,H2) = 0 D(K1,K2) = −2.75 D(L1, L2) = −4H1 ≺ H2 K1 ≺ K2 L1 ≺ L2

Example 4.5. Consider the triangular fuzzy number A = (1, 2, 5), and the general numberB, shown in Fig. 14., that the membership function of B is defined by

B(x) =

√1− (x− 2)2 when x ∈ [1, 2],

√1− 1

4(x− 2)2 when x ∈ [2, 4],

0 otherwise.

In Zhong-xing Wang and Ya-ni Mo [17] ranking method, there was C∗A = 0.4933 and C∗

B =0.5067, therefore A ≺ B. By using proposed method, there is D(A, B) = 2− 4.1886 = −2.1886.Thus, the ranking order is A ≺ B. From Fig. 14, we can conclude that A ≺ B is more consistentwith human intuition.


Fig. 14.

5. Some properties

In this section, some valuable properties which are useful in ranking a large quantity offuzzy numbers simultaneously are described. Assume that there are m different bounded fuzzynumbers, A1, A2, . . . Am, to be ranked. Let Ai, Aj , Ak be any three arbitrary fuzzy numbers,where i 6= j 6= k and 1 ≤ i, j, k ≤ m

(1) D(Ai, Ai) = 0.

(2) D(Ai, Aj) = −D(Aj , Ai).

(3) If D(Ai, Aj) > 0 and D(Aj , Ak) > 0, then D(Ai, Ak) > 0.

Therefore, if Ai > Aj and Aj > Ak are known, we can infer that Ai > Ak. In fact, thetotal difference among three fuzzy numbers has the following relation: D(Ai, Ak) = D(Ai, Aj)+D(Aj , Ak).

In other words, once the values of D(Ai, Aj) and D(Aj , Ak) are known, the value of D(Ai, Ak)is determined by simple arithmetic computation.

(4) More than two fuzzy numbers can be ranked by comparing the benchmark fuzzy number.Let Aj be the benchmark, and D(Ai, Aj) = a and D(Ak, Aj) = b. By using the previous twoproperties, obviously D(Ai, Ak) = D(Ai, Aj) − D(Ak, Aj) = a − b. Therefore, if a > b, thenD(Ai, Ak) > 0; i.e. Ai > Ak.

(5) If D(Ai, Aj) < ε, then |D(Ai, Ak)−D(Aj , Ak)| < ε.

This equation suggests that the total difference between one fuzzy number and the other twofuzzy numbers are insignificant if the two fuzzy numbers are close to each other. This propertyholds, since D(Ai, Ak) = D(Ai, Aj) + D(Aj , Ak). When using the above properties, for rankingm fuzzy numbers, only m−1 comparisons to the benchmark fuzzy number are necessary, insteadof m(m− 1)/2 or m comparisons. For example, letting Aj be the benchmark, then only m− 1values of the total difference, D(A1, Aj), D(A2, Aj), ..., D(Aj−1, Aj), D(Aj+1, Aj), ..., D(Am, Aj)are necessarily determined.

(6) This method can compare nonnormal fuzzy numbers. Thus the proposed approach is moreefficient than the existing ranking methods in [7].


This article considers the following reasonable axioms that Wang and Kerre [23] proposed forranking fuzzy quantities.

Let D be an ordering method, S the set of fuzzy quantities for which the method D can beapplied, and A a finite subset of S. The statement ”two elements A and B in A satisfy that A

has a higher ranking than B when D is applied to the fuzzy quantities in A” will be written as”A Â B by D on A”. ”A ∼ B by D on A”, and ”A º B by D on A” are similarly interpreted.The axioms as the reasonable properties of ordering fuzzy quantities for an ordering approachD are as follows [23]:A-1. For an arbitrary finite subset A of S and A ∈A; A º A by D on A.A-2. For an arbitrary finite subset A of S and (A,B) ∈ A2; A º B and B º A by D on A, weshould have A ∼ B by D on A.A-3. For an arbitrary finite subset A of S and (A,B, C) ∈ A3; A º B and B º C by D on A,we should have A º C by D on A.A-4. Let S and S

′be two arbitrary finite sets of fuzzy quantities in which D can be applied and

A and B are in S ∩ S′. We obtain the ranking order A º B by D on S

′iff A º B by D on S.

A-5. Let A, B, A + C and B + C be elements of S. If A º B by D on A and B, thenA + C º B + C by D on A + C and B + C.A′-5. Let A, B, A + C and B + C be elements of S. If A Â B by D on A and B, then

A + C Â B + C by D on A + C and B + C.A-6. Let A, B, AC and BC be elements of S and C ≥ 0. A º B by M on {A,B} impliesAC º BC by M on {AC, BC}.

The easy proofs of the properties mentioned above are left to the reader.

6. Conclusion

For ranking a large quantity of fuzzy numbers based on only limited information aboutthem, an effective, efficient, and accurate ranking method becomes necessary. Most of methodswhich were proposed up to now have pitfalls in some aspects, such as inconsistency with humanintuition and indiscrimination. In proposed method, the core and margin and α − cuts havebeen considered corresponding to each of fuzzy numbers simultaneously. Besides, all fuzzynumbers can be compared by our method, except real numbers. Also in this paper we describedsome properties which are useful in ranking a large quantity of fuzzy numbers. With respectto the examples, it is obvious that the proposed method is more reliable and computationallyeasier than some methods which were proposed up to now and overcome the shortcoming ofthe previous methods. Also, the proposed method can rank normal/nonnormal triangular andtrapezoidal fuzzy numbers, but some methods, which were proposed up to now, were not ableto do this.

7. Acknowledgements

We would like to thank the anonymous referees for their careful corrections and valuablesuggestions for improving the paper.

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Reza Ezzati was born in Tabriz, Iran in 1974.

He received his B.Sc. degree in pure mathematics

in 1997 and M.Sc. degree in applied mathematics

from Azerbaijan Teacher Education University and

Tarbyat Modarres University of Tehran, respectively,

and his Ph.D. degree in applied mathematics from Is-

lamic Azad University, Science and Research Branch,

Tehran, Iran in 2006. He is an assistant professor in

the Department of Mathematics at Islamic Azad Uni-

versity, Karaj Branch, (Iran). His current interests

include fuzzy mathematics, especially, on numerical

solution of fuzzy systems and fuzzy interpolation, ite-

rative methods for solving nonlinear equations.

Ramin Enayati was born in Tehran, Iran, in 1982.

He received the M.S. degree in applied mathematics

from Department of Mathematics, Islamic Azad Uni-

versity, Science and Research Branch, Tehran, Iran

in 2008. Since 2009, he is Ph.D. research student

in Operation Research at Islamic Azad University,

Karaj, Iran. His research interests include operation

research, MOLP problems and fuzzy logic.

Azam Mottaghi was born in Iran in 1983. She re-

ceived M.Sc. from Islamic Azad University, Mashhad

Branch in 2008. She is currently doing her Ph.D re-

search under supervision of Dr. Esmaeil Khorram and

Dr. Reza Ezzati at the Department of Mathematics,

Islamic Azad University, Karaj Branch. Her research

interests include multi objective programming with

fuzzy variable, nonlinear programming problem.


Rahim Saneifard was born in Oroumieh, Iran in

1972. He received B.Sc. in pure mathematics in 1997

and M.Sc. in applied mathematics from Azerbaijan

Teacher Education University in Tabriz, Islamic Azad

University and from Lahijan Branch in Lahijan, re-

spectively. He is a Ph.D. student in the Department

of Mathematics at Islamic Azad University, Science

and Research Branch, Tehran, in Iran. His current

interest include fuzzy mathematics.

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